check out the media option on cascading stylesheets.
good links include:
http://alistapart.com/stories/goingtoprint/
http://www.richinstyle.com/
essentially it's saying you have your normal "screen" stylesheet and you have another when it's printed.
So a class called .header might be "visible" on the screen but "hidden" in print.
Play around, you will need to consider your design carefully but if you plan well it shouldn't be a problems.